初步探讨了D. H.莱默猜想
An elementary approach to toy models for D. H. Lehmer's conjecture
摘要：
在1947年，莱默证明，拉马努金的tau函数$τ(m)$永远不会消失的所有正整数$m$，其中$τ(m)$是$m$ -th傅立叶系数的尖点形式$Δ_{24}$的重量12。球形$t$ -设计理论与Lehmer猜想密切相关，因为Venkov，de la Harpe和Pache证明了$τ(m)=0$等价于$E_{8}$ -格的范数$2m$的壳是球形8-设计。因此，Lehmer猜想在球形$t$ -设计中被重新表述. Lehmer的猜想很难证明，仍然是开放的。然而，Bannai-Miezaki证明了$\RR^2$中整数格$\ZZ^2$的非空壳都不是球面4-设计，六边形格$A_2$的非空壳都不是球面6-设计。此外，与类数为1或2的虚二次域的代数整数相关联的整数格的非空壳，除了$\QQ(\sqrt{-1})$和$\QQ(\sqrt{-3})$之外，都不是球面2-设计.在证明中，模形式理论起了重要作用。最近，Yudin发现了一个关于$\ZZ^{2}$ -格的初等证明，它不使用模形式的理论，而使用Calcut的最新结果。在本文中，我们给出了基本的（即，模形式自由）的证明，并讨论了Calcut的结果与虚二次域理论之间的关系。
英文摘要：
In 1947, Lehmer conjectured that the Ramanujan's tau function $τ(m)$ never vanishes for all positive integers $m$, where $τ(m)$ is the $m$-th Fourier coefficient of the cusp form $Δ_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $τ(m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice $\ZZ^2$ in $\RR^2$ is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for $\QQ(\sqrt{-1})$ and $\QQ(\sqrt{-3})$ is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of $\ZZ^{2}$-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
